Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding oriented matroid MΦ, then B (MΦ) is dual ...

We consider an extension to the optimal matroid base problem [1] whereby the matroid element costs are not fixed, but are time dependent. We propose a genetic algorithm (GA) approach to solve the optimal sequenced matroid base problem (OSMBP) by employing efficient codes which are suffixed by a standard permutation code [2]. These novel encoding schemes insure feasibility after performing the c...

In this paper we highlight some enumerative results concerning matroids of low rank and prove the tail-ends of various sequences involving the number of matroids on a finite set to be log-convex. We give a recursion for a new, slightly improved, lower bound on the number of rank-r matroids on n elements when n = 2 − 1. We also prove an adjacent result showing the point-lines-planes conjecture t...

A signed graph is a representation of an even cycle matroid M if the cycles of M correspond to the even cycles of that signed graph. Two, long standing, open questions regarding even cycle matroids are the problem finding an excluded minor characterization and the problem of efficiently recognizing this class of matroids. Progress on these problems has been hampered by the fact that even cycle ...

This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M , (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)| ≥ 4r(M)− 5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M) ≥ 4 and M has a basis B such that M\e is not 3-connected for al...

We prove a natural bijection between the polytopal tilings of a zonotope Z by zonotopes, and the one-element-liftings of the oriented matroid M(Z) associated with Z. This yields a simple proof and a strengthening of the Bohne-Dress Theorem on zonotopal tilings. Furthermore we prove that not every oriented matroid can be represented by a zonotopal tiling.

We give a characterization of the internally 4-connected binary matroids that have no minor isomorphic to M(K3,3). Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Möbius ladder, or is isomorphic to one of eighteen sporadic matroids. 2000 Mathematics Subject Classification. 05B35.

The main purpose of this paper is to introduce a concept of$L$-fuzzifying topological groups (here $L$ is a completelydistributive lattice) and discuss some of their basic properties andthe structures. We prove that its corresponding $L$-fuzzifyingneighborhood structure is translation invariant. A characterizationof such topological groups in terms of the corresponding$L$-fuzzifying neighborhoo...

provided again that M is simple and binary. The reason for restricting attention here to simple binary matroids is that expression (1.1) need not hold for arbitrary simple matroids or even for loopless graphic matroids. To see, this, consider, for example, the uniform matroids U 2 ,m form~ 6 and U 1,n for n ~ 4. We observe here that inequality (1.3) is a sharper bound than expression (1.2) unle...

We consider the problem of determining precisely which graphs G have the property that the split-off operation, by every pair of edges, on the cycle matroid M(G) yields a graphic matroid. This problem is solved by proving that there are exactly four minor-minimal graphs that do not have this property.